Subject: Modeling of Thermal Rocket Engines; Nozzle flow; Control of mass flow. p c. Thrust Chamber mixing and combustion

Transcription

1 16.50 Leture 6 Subjet: Modeling of Thermal Roket Engines; Nozzle flow; Control of mass flow Though onetually simle, a roket engine is in fat hysially a very omlex devie and diffiult to reresent quantitatively by mathematial models. But the diffiulty and high ost of arriving at a suessful design makes it essential that the erformane-limiting henomena be desribed as aurately as ossible. It is no longer feasible to aroah the develoment of a new roket engine by a urely ut-and-try method. The range of hemomena that must be dealt with is suggested by the sketh. They range from the fluid dynamis of the uming system through the very omlex henomena of ombustion to strutures with heat transfer rates beyond those exeriened anywhere other than in reentry from sae. Hot gas, T 3D Suersoni Flow Pum e u e Fuel Oxidizer Thrust Chamber mixing and ombustion q Heat Transfer To deal with this omlex situation, we use models that reresent adequately the henomena we want to deal with, while suressing other henomena. Some models we will find useful are those for: 1) Turboums 2) Injetors 3) Combus ion 4) Nozzle Flow 5) Heat Transfer 6) Struture of ombustion hambers and nozzles 1

2 Nozzle Flow Let us fous on the Nozzle Flow first. For this urose we regard the ombustor as just a soure of hot gas of known hemial omosition, stagnation temerature T and stagnation ressure. We have already used a very simle model to reresent the result of this flow roess. Assuming that the exansion was to zero ressure and therefore zero temerature we used onservation of energy to show that = 2 T = 2! RT (model 1)! "1 where was defined as the "exhaust veloity" meaning the veloity whih, multilied by the mass flow rate, gives the thrust. Now we want to exlore the fluid mehanial roesses in more detail, and also reresent the harateristis of the gas more aurately. The gas is in fat a mixture of gases resulting from the ombustion roess Its omosition is seified in terms of mole frations y i of seies i or mass frations x i of seies i. For examle if the roket burns H 2 and O 2 in a mixture ratio of 3 moles of H 2 to 1 mole of O 2 to rodue H 2 0 and H 2, (by mass O/F=32/(3x2)=5.3, omarable to O/F=6 in the Shuttle SSME engine), 3H ! 2H H 2 then the mole frations and mass frations of H 2 0 and H 2 are as shown in the table. H 2 H 2 0 x 1/19 18/19 y 1/3 2/3 With these mole or mass frations we an omute the roerties of the gas mixture, suh as the seifi heats, the gas onstant and γ. The fluid-mehanial and heat transfer henomena that are of imortane to desribe are: 1) The effets of nozzle shae on flow 2) The effets of hemial reations during the flow 3) The heat transfer from the flow to the nozzle 2

3 Effet of Nozzle Shae The henomena that influene the ouling between nozzle shae and the flow an be divided into two general lasses, first those assoiated with the gas dynamis and seond those assoiated with the roerties of the gas. Deending on how aurately we attemt to model eah of these, our model of the flow an vary enormously in omlexity. The range of ossibilities is shown shematially in the figure, where sohistiation of the modeling inreases away from the origin along two axes, one measuring gas dynamis, the other fluid roerties. MOC with ideal gas CFD+Combustion Dusty flow Frozen, but (T) One-D reating gas Along the gas dynami axis the desrition an range from the simle hannel flow model that we will disuss here to very sohistiated three-dimensional desritions of the transoni flow. Similarly along the roerties axis the range is from the thermally and alorially erfet gas model we will use (often alled the "ideal gas") to full kineti desrition of the effets of hemial reations that our as the gas flows through the nozzle. The simlest model of all (Model 1) equates the kineti energy of the exhaust flow to the thermal energy in the hamber Here we will limit the disussion of the ouling of flow to the geometry of the nozzle to a simle model (Model 2) at the origin. The imliations of the more omlex henomena will then be disussed searately through two more models. Model 5 will deal with the effets of hemial reations, and Model 3 with the effets of nozzle shae other than the simle area variation. Model 4 will reresent some seial hemial and fluid mehanial asets of solid roket motors. 3

4 Model 2: Channel Flow of Perfet Gas The hannel flow model (often erroneously referred to as "one-dimensional flow") is one in whih the entire flow is onsidered to be in a single stream tube having a mass flow that is onstant along the axis of the stream tube. The stream tube is onsidered to be narrow enough that the angle of the walls to the axis may be onsidered small, the gas roerties nearly onstant aross the stream tube, and the omonents of veloity normal to the axis small enough that their squares are negligible omared to that of the streamwise veloity. If the ross-setional area of the streamtube is denoted A(x), the onservation of mass beomes ρua(x) = ṁ = onstant (1) The onservation of energy beomes h + = h (Note: v2 <<1) 2 u 2 or if the seifi heat is assumed onstant. u 2 u2 T + = 2 T (2) The final relationshi defining the flow is a onnetion between two of the thermodynami roerties of the fluid. In general this is a desrition of the variation of the entroy of the fluid along the flow diretion. For our simle model we assume that the flow is Isentroi (Adiabati and Reversible). The entroy is referred to a referene state whih we take to be the hamber ondition. Conventionally, we assume that the veloity in the hamber is low enough that this is also the stagnation ondition in the hamber. T s! s = " dt! R!n = 0 T T so if s is onstant, and is as well,!n T! R!n = 0 and from this it follows that T T R = ( ) C! "1 = ( )! T (3) where! " ; - v =R v We want to maniulate these three relationshis to arrive at an exression giving the variation of Mah number with A, hene with x. We begin by exressing equation (2 )in the form: T u 2! " 1 u 2 = 1 + = 1 + M 2, M 2! T 2 T 2 "RT 4

5 and then from (3)! "1 = (1 + M 2 )! "1 2 With these we an write (1) in the form, T 1 T ṁ =!ua = ua = M "RT RT T R! 1! "1 2! "1 = (1 + M 2 ) " + M!T / RA 2 T ṁ! M = A RT 1 (1 +! " M 2 ) 2! +1 2(! "1) For given (and onstant) and T, this exression gives the mass flow er unit area as a funtion of the Mah number M. Sine the mass flow is onstant, we have the desired exression for M as a funtion of A(x). We see that the mass flow er unit area, ρu, is roortional to the stagnation ressure and inversely roortional to the square root of the stagnation temerature. It is both onventional and useful to lot the funtion of Mah number in ratio to its value at the oint in the flow where M=1. The value of ρu at this oint we all ρu*. It is given by " + 1 " # " +1 ' 2 (!u)* % (" '1) = RT $ 2 & Then the ratio is 2(" #1) $ " + 1 '!u A * & ) = = M 2 & * 1 )!u A & 1 + " # M 2 ) % 2 ( This funtion of M is shown in the figure. C! " +1 T T 5

6 Some of the dominant features are: 1) There is a maximum in the mass flow er unit area at M=1. 2) For M<1, ρu (and also u) inreases with M, so M inreases with dereasing A. 3 For M>1, ρu (and also u) dereases with M, so M inreases with inreasing A. We note however that always dereases with inreasing M. It follows that if we want to inrease M from a small value to one in exess of 1, as we do in a roket nozzle, the nozzle should be shaed as in the sketh at the beginning of the leture, with a onvergene to a minimum area "throat", followed by a divergene. For given and T, the area of the throat ontrols the mass flow. To get an idea of the required throat size we an evaluate the (dimensional) fator in front of the funtion of M for a set of tyial values. Say = 100 atm = 10 7 N/m 2 T = 3000 K! = 1.2 R = 8.32x10 3 = 640 J/kg/K 13! 1.2 =10 7 = 7.91x10 3 kg /m 2 /s RT (640)(3000) We define the throat of the nozzle as the oint of minimum area, where M=1 and all the area there A t. Then 6

7 ṁ! 2 = ( ) A t RT! +1)! +1 2(! "1) The onlusion that the oint of minimum area in the nozzle ontrols the mass flow is valid beyond the limits of our simle ideal gas mode.. More generally, we define a quantity * by ṁ! A t that is a measure of the mass flow er unit area at the throat. For the ideal gas * model, RT 1! +1! + 2(! "1) * = ( )! 2 For the examle above, (640) * = ( ) 5.5! 2138m / s In general * deends only on the roellant ombination and. One further ondensation of these formulas that is sometimes useful is to define the " +1 2 quantity! = " ( ) 2(" #1), whih is lose to 0.65 for most values of γ, and then write " +1 RT * = ; m = A t! * 7

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